Natural Units - Choosing Constants To Normalize

Choosing Constants To Normalize

Out of the many physical constants, the designer of a system of natural unit systems must choose a few of these constants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the mass of a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, then the mass of a proton has to be ≈1836. In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to 1, because it is a dimensionless number. The fine-structure constant is related to other fundamental constants

where k_e is the Coulomb constant, e is the elementary charge, ℏ is the reduced Planck constant, and c is the speed of light. Therefore it is not possible to simultaneously normalize all four of the constants c, ℏ, e, and k_e.